Bouncers, or nightclub security, do not have the same rights as the police or actual security personnel. They do not have the right to physically remove a person from the premises of the club, but this is a very common misconception. Bouncers do not have the legal right to engage in violence or force as they see fit. In general, bouncers are regular employees, not actual security personnel. Actual security personnel require special training and permits.
Bouncers are only allowed to use force if it is first used against them. Therefore, according to law, a bouncer may only ask you to leave the premises. They do not have the right to physically remove a customer. These same rights to self defense are granted to ordinary citizens, as is the right to call the police on someone.